I think the main disadvantage is the expected end-user experience.
I think this might be true for poor implementations of Ajax. But if used correctly, Ajax should enhance the user experience. In most cases, these enhancements don't even need to be all that noticeable to the user. Examples of a good Ajax implementation would be refreshing part of the page instead of the whole thing. A bad implementation would be a type-ahead feature that slows the data entry process down rather than speeding it up.
So in my mind, the disadvantages of Ajax are the same as those of any web technology... it needs to be implemented judiciously, with the user's experience in mind.